Cognition and Incentives in Cooperatives

Abstract

We extend the results of Feng and Hendrikse (2012) by investigating the relationship between cognition and incentives in cooperatives versus investor-owned firms (IOFs) in a multi-tasking principal-agent model. The principal chooses the incentive intensity as well as the precision of monitoring, while the agent chooses the activities. We establish that a cooperative is uniquely efficient when either the synergy between the upstream and downstream activities or the knowledgeability of the members regarding the cooperative enterprise is sufficiently high.

Appendix

A. Total Surplus Figures

The figures representing the total surplus of the upstream IOF, the downstream IOF, and the cooperative, respectively (Fig. 7):

Fig. 7

The total surplus of the upstream IOF, the downstream IOF, and the cooperative

B. Proofs of Comparative Statics Results

Denote $\left[\left({f}_u{g}_u-k{f}_d{g}_u\right)-\sqrt{2r}\left(1-{k}^2\right)W\right]$ as F > 0.

$$ \frac{\partial {\beta}^{\ast }}{\partial k}=\frac{1}{g_u^2}\left(2\sqrt{2r} kW-{f}_d{g}_u\right)<0;\frac{\partial {\beta}^{\ast }}{\partial {\omega}_{ii}}=-\frac{1}{g_u^2}\sqrt{2r}\left(1-{k}^2\right)<0; $$

$$ \frac{\partial {V}_u^{\ast }}{\partial k}=-\frac{\sqrt{\frac{2{\omega}_{uu}}{r}}{g}_u^2\left(2\sqrt{2r} kW-{f}_d{g}_u\right)}{F^2}>0;\frac{\partial {V}_u^{\ast }}{\partial {\omega}_{iu}}=\frac{\left[F\sqrt{\frac{1}{2r{\omega}_{iu}}}{g}_u^2+{g}_u^2\left(1-{k}^2\right)\right]}{F^2}>0; $$

$$ \frac{\partial {V}_u^{\ast }}{\partial {\omega}_{id}}=\sqrt{\frac{r}{2{\omega}_{id}}}\left(1-{k}^2\right)/{F}^2>0; $$

So as

$$ \frac{\partial {V}_d^{\ast }}{\partial k}>0,\frac{\partial {V}_d^{\ast }}{\partial {\omega}_{iu}}>0,\frac{\partial {V}_d^{\ast }}{\partial {\omega}_{id}}>0; $$

In terms of $\frac{\partial E\left(\pi +U\right)}{\partial k}$, for −1 < k < 0 and when k is increasing, (f_u − kf_d) is decreasing, $\sqrt{2\left(1-{k}^2\right)}$ is increasing, $\sqrt{r\left(1-{k}^2\right)}$ is increasing, so the overall effect is decreasing, $\frac{\partial E\left(\pi +U\right)}{\partial k}<0$; denote $\left(\frac{f_u-k{f}_d}{\sqrt{2\left(1-{k}^2\right)}}-\frac{\sqrt{r\left(1-{k}^2\right)}}{g_u}W\right)$ as T,

$$ \frac{\partial E\left(\pi +U\right)}{\partial {\omega}_{iu}}=\frac{-\sqrt{\frac{r\left(1-{k}^2\right)}{\omega_{iu}}}}{g_u}T<0,\frac{\partial E\left(\pi +U\right)}{\partial {\omega}_{id}}=\frac{-\sqrt{\frac{r\left(1-{k}^2\right)}{\omega_{id}}}}{g_u}T<0. $$

C. Efficiency Numerical Illustrations

We would like to investigate under which condition the difference between the cooperative’s total surplus to the sum of the two IOFs can be positive (indicating the cooperative is efficient). Here we denote H as the difference, and H has the form below

$$ H={\left(\frac{f_u-k{f}_d}{\sqrt{2\left(1-{k}^2\right)}}-\frac{\sqrt{r\left(1-{k}^2\right)}}{g_u}\left(\sqrt{\omega_{uu}}+\sqrt{\omega_{ud}}\right)\right)}^2-{\left(\frac{f_u}{\sqrt{2\left(1-{k}^2\right)}}-\frac{\sqrt{r\left(1-{k}^2\right)}}{g_u}\sqrt{\omega_{uu}}\right)}^2-{\left(\frac{f_d}{\sqrt{2\left(1-{k}^2\right)}}-\frac{\sqrt{r\left(1-{k}^2\right)}}{g_d}\sqrt{\omega_{dd}}\right)}^2 $$

To solve for H > 0, we will have a function describing the relations between ω_ud and k.

First, we directly calculate the above function and report the result from Matlab. ω_ud(k) should be in the following interval to make the cooperative efficient:

$$ {\displaystyle \begin{array}{l}\Big(-\frac{\sigma_2- ab+\sqrt{2} ab+ abk-{\sigma}_1}{\sigma 3},\\ {}-\frac{\sigma_2- ab-\sqrt{2} ab+ abk+{\sigma}_1}{\sigma 3}\Big)\end{array}} $$

$\mathrm{where}\ {\sigma}_1=\sqrt{2}\;w\;\sqrt{2-2\;{k}^2}\;{\sigma}_4;{\sigma}_2=w\;\sqrt{2-2\;{k}^2}\;{\sigma}_4;{\sigma}_3=\sqrt{2-2\;{k}^2}\;{\sigma}_4$; ${\sigma}_4=\sqrt{-r\;\left({k}^2-1\right)}$; a and b are functions of the productivity and performance measurement parameters.

The explicit result is too lengthy to analyze. Two numerical examples are presented to illustrate various insights. The first example demonstrates that stronger cognition capacity of the upstream party offsets the negative effects of lower chain interdependency, allowing cooperatives to outperform the IOFs; the second example shows that when cognitive capacity is low, stronger interdependency within the industry still allows cooperatives to be more efficient (Fig. 8).

Fig. 8

Two examples of how cognition capacity and chain interdependency can complement each other and make the cooperative more effective

We set values for variables for both examples as follows: ${f}_u={f}_d=20;{g}_u={g}_d=1;{\omega}_{uu}={\omega}_{dd}=4;r=0.5,\frac{f_u-\sqrt{f_u^2+{f}_d^2}}{f_d}=1-\sqrt{2}$.

For point A, we set ω_ud = 0.16 < ω_uu = ω_dd, and $k=-0.4>\frac{f_u-\sqrt{f_u^2+{f}_d^2}}{f_d}$. H = 2.34 > 0. Table 1 below shows the values of other variables.

Table 1 The parameter values of point A showing that the cooperative can be efficient even when the chain synergies are low

The values in the second example, determining point B, are ω_ud = 9 > ω_uu = ω_dd, and $k=-0.5<\frac{f_u-\sqrt{f_u^2+{f}_d^2}}{f_d}$. H = 3.04 > 0. ($\sqrt{\omega_{ud}}>\frac{g_u}{\sqrt{2r}\left(1-{k}^2\right)}\left[\left({f}_u-k{f}_d\right)-\sqrt{f_u^2+{f}_d^2}\right]-\sqrt{\omega_{uu}},$ so B is above the curve.). Table 2 below shows the values of other variables.

Table 2 The parameter values of point B showing that the scope for the efficiency of the cooperative increases when the cognitive capacities of the downstream party decrease

A is an efficient point for the cooperative, even though the chain interdependency is not so high. This is evidence of the cognition advantage of the cooperative. Without introducing the monitoring intensity principle, efficient point A would never be accessible. But a low enough ω_ud (indicating a high enough cognitive ability, due to innovation and information sharing for instance) can compensate such a high k, leading the cooperative to be efficient in the end.

B is another efficient point for the cooperative, even with a much lower cognitive ability. It justifies the interdependency advantages of the cooperative. In an environment that the processor of the cooperative knows so little, but the chain interdependency is high enough to cover, then the IOFs can still be dominated.

Here we would like to highlight the mechanism behind. Recall that the expected value of the total surplus is the sum of the principal’s payoff and CEO’s payoff: $E\left(\pi +U\right)=y-c\left({a}_{iu},{a}_{id}\right)-\frac{1}{2}r\mathrm{Var}(w)-M(V)$. In the first example, a higher k directly increases the costs of the activities and indirectly drops the activities, causing the total output y to decrease, but a much lower ω_ud reduces the monitoring costs and thus the risk aversions more compared to the IOFs, leading to an efficient cooperative. In Table 1, the sum of the cooperative activities is slightly less than the IOFs’ activities, but the sum of the variances is just the opposite, especially for the downstream variance. This justifies that the cognition effect matters more in this example, and by reducing the variances-related costs, the cooperative is performing better than the two IOFs.

The other example is just the other way around. When ω_ud increases, the monitoring costs and risk aversions rise, but a lower k will trigger higher activities and output to compensate more. In Table 2, the variances are kind of similar, but the sum of the cooperative activities is larger than the IOFs’ activities. It verifies that the chain interdependency effect counts more in this case, and by motivating more activities, the cooperative can be efficient.